Optimal. Leaf size=20 \[ \sin ^{-1}\left (\frac{x}{2}\right )-\sqrt{4-x^2} \]
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Rubi [A] time = 0.0048059, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {641, 216} \[ \sin ^{-1}\left (\frac{x}{2}\right )-\sqrt{4-x^2} \]
Antiderivative was successfully verified.
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Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{1+x}{\sqrt{4-x^2}} \, dx &=-\sqrt{4-x^2}+\int \frac{1}{\sqrt{4-x^2}} \, dx\\ &=-\sqrt{4-x^2}+\sin ^{-1}\left (\frac{x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.011744, size = 20, normalized size = 1. \[ \sin ^{-1}\left (\frac{x}{2}\right )-\sqrt{4-x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 17, normalized size = 0.9 \begin{align*} \arcsin \left ({\frac{x}{2}} \right ) -\sqrt{-{x}^{2}+4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.72742, size = 22, normalized size = 1.1 \begin{align*} -\sqrt{-x^{2} + 4} + \arcsin \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82944, size = 70, normalized size = 3.5 \begin{align*} -\sqrt{-x^{2} + 4} - 2 \, \arctan \left (\frac{\sqrt{-x^{2} + 4} - 2}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.144732, size = 12, normalized size = 0.6 \begin{align*} - \sqrt{4 - x^{2}} + \operatorname{asin}{\left (\frac{x}{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.68049, size = 22, normalized size = 1.1 \begin{align*} -\sqrt{-x^{2} + 4} + \arcsin \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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